Best leather for wallets: bridle hide? calf skin? alligator?

I guess that largely depends on what you want and what the wallet should look like. Shell cordovan and boxcalf leather works out really well wheras bridle leather is really durable. Most of all the craftsmanship employed outweighs the importance of the leather although it is mostly a matter of price to find the perfect combination of both. Good leather will make your wallet beautiful for a longer time, but good craftsmanship is important that the substance is well made. If the edges and the stitching is not symmetric and solidly made even the most beautiful leather loses its appeal. 

Bridle leather is termed as such due to the processes it goes through during the tanning process. The bottom level of bridle leather is very high in term of quality whereas calf, gator/croc and others only describe the animal from which it came, not the process it underwent.

So the quality is very variable with the others. Bridle leather is also meant specifically for heavy use, although I will say that most wallets employ much thinner leather of any given type, otherwise you would have a 3 inch tall wallet. At the end of the day, if you find a wallet in any full grain leather or real gator/croc, go for it.

This also accounts for shell Cordovan from Horween where the quality is guaranteed. Bridle leather is usually very heavy which makes it difficult to use for wallets. Alligator skin is very thin and easier to use as is calf skin.